Prepare a medium-high gas or charcoal grill fire.
Melt the butter in a 12-inch skillet over medium-low heat. Add the onions and cook, stirring often, until golden and soft, about 20 minutes; reduce the heat to low if the onions begin to brown too quickly.
Meanwhile, in a large bowl, thoroughly mix the beef, tarragon, mustard, and Worcestershire sauce with your hands. Form the mixture into four 1/2-inch-thick patties.
Oil the grill grate. Grill the burgers, covered, for 3 minutes. Flip the burgers and top with the cheese. Cover and continue to cook until an instant-read thermometer inserted into a burger registers 130°F for medium rare, about 4 minutes more, or 145°F for medium, about 6 minutes more.
When the burgers are nearly done, toast the rolls cut side down on the grill until browned and heated through, 1 to 2 minutes. Serve the burgers in the rolls, topped with the caramelized onions.